第十二讲—空间群(4)

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,第十一讲,空间群,(3),：,非点式空间群,1,、,非点式空间群,举例分析,2,、,空间群国际表,举例分析,3,、,二维空间群,(,全部,),第十二讲,空间群,(4),：,空间群中的特殊位置,1,、,空间群国际表中的特殊位置,举例分析,2,、,二维空间群,(,全部,),P,4nc,4mm Tetragonal,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,8,c,1,x,y,z,;,x,y,z,;,+x,-y,+z,;,-x,+y,+z,;,y,x,z,;,y,x,z,;,+y,+x,+z,;,-y,-x,-z,.,4,b,2,0,z,; ,0,z,;,0,+z;,0,+z.,2,a,4,0,0,z,; , , +z.,Origin on 4,Number of positions,Wyckoff notation,and point symmetry,Co-ordinates of equivalent positions,6,C,4v,No.,104,P,4nc,Conditions limiting possible reflections,General:,hkl,: No conditions,0,kl,:,k,+,l,= 2n,hkl,:,l,= 2n,Special:,hkl,:,h,+,k,= 2n;,l,= 2n,hkl,:,h,+,k,+,l,= 2n,+,+,+,+,1,a,1,x,y,z,2,i,1,x,y,z,;,x,y,z,.,1,h,1,0,.,1,g,1,0,.,1,f,1,0,.,1,e,1,0.,1,d,1,0,0.,1,c,1,0,0.,1,b,1,0,0,.,1,a,1,0,0,0.,+,_,+,_,+,_,+,_,Wyckoff,符号,P,1(C,1,No,. 1),1,P,1(C,i, No. 2),1,Origin at 1,等效位置数,特殊位置,位置对称性,F,m3m (O,h,No,. 225),5,I,m3m (O,h,No,. 229),9,192,l,1,x,y,z,;,.,.,8,c,4,3m,; ,.,4,b,m3m, .,4,a,m3m,0,0,0.,(0,0,0; 0,; ,0,; ,0) +,96,l,1,x,y,z,;,.,.,8,c,3m,; ,; ,; ,.,6,b,4,/mmm,0,; ,0,; ,0.,2,a,m3m,0,0,0.,(0,0,0; ,) +,点式空间群：,由全部作用于同一个公共点上的对称操作完全确定，或者说仅由点对称操作和平移对称操作组合而产生。,螺旋轴或滑移面不是其,基本操作,。,至少有一个,基本操作,为非点式操作即是非点式空间群,点式空间群在单胞中一定至少有一个位置具有与空间群点群相同的,位置对称性,空间群操作：,r,= ,R,|,t,r,=,R,r,+,t,(,赛兹算符,),对非点式操作,t,=,，,是单胞的分数平移,对于点式操作,t,=,=,0,R,|,t,、,1,|,t,n,、,R,|,0,、,R,|,一般来说，对于给定的一组等效位置，,等效位置数,乘以,位置对称性,点群的阶,等于,空间群点群的阶,P,62m (D,3h,No,. 189),3,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,+,+,_,_,Origin at 62m,12,l,1,x,y,z,;,y,x-y,z,;,y-x,x,z,;,x,y,z,;,y,x-y,z,;,y-x,x,z,;,y,x,z,;,x,y-x,z,;,x-y,y,z,;,y,x,z,;,x,y-x,z,;,x-y,y,z,.,1,b,62m,0,0,.,1,a,62m,0,0,0,.,Wyckoff,符号,P2,+,+,+,+,+,+,+,+,2,e,1,x,y,z,;,x,y,z,.,1,d,2, , z.,1,c,2, 0, z.,1,b,2,0, , z.,1,a,2,0, 0, z.,P2,1,+,+,+,+,+,+,+,+,2,a,1,x,y,z,; x,y,+z.,Pm,+,-,+,-,+,-,+,-,-,+,+,-,+,+,Pb,2,c,1,x,y,z,;,x,y,z,.,1,b,m,x, y, .,1,a,m,x, y, 0.,2,a,1,x,y,z,; x,+y,z.,特征,1,：特殊位置,与,非点式操作,Origin on 2,+,+,_,_,+,_,+,_,+,+,_,_,+,_,+,_,+,+,_,_,+,_,+,_,+,+,_,_,+,_,+,_,8,p,1,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,y,x,z,;,y,x,z,;,y,x,z,;,y,x,z,.,1,a,42,0,0,0,.,P,422 (D,4,No. 89,),1,P,42,1,2 (D,4,No. 90,),2,8,g,1,x,y,z,;,x,y,z,;,-x,+y,z,;,+x,-y,z,;,y,x,z,;,y,x,z,;,+y,-x,z,;,-y,+x,z,.,2,a,222,0,0,0,; ,0.,特征,2,：,点式,SG,与,非点式,SG,+,_,_,+,+,_,+,_,+,_,_,+,+,_,_,+,+,_,_,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,P,mm2 (C,2v,No,. 25),1,Origin on mm2,4,i,1,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,.,2,h,m,y,z; ,y,z.,2,g,m,0,y,z; 0,y,z,2,f,m,x,z; x,z.,2,e,m,x,0,z; x,0,z,1,d,mm,z.,1,c,mm,0,z.,1,b,mm,0,z.,1,a,mm,0,0,z.,P,cc2 (C,2v,No,. 27),3,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,4,e,1,x,y,z,;,x,y,z,; x,y,+z; x,y,+z.,2,d,2,z; ,+z.,2,c,2,0,z; ,0,+z,2,b,2,0,z; 0,+z.,2,a,2,0,0,z; 0,0,+z,Origin on 2,P,ban,(,D,2h,No,. 50),4,P,2/,b,2/,a,2/,n,+,+,-,-,-,-,+,+,-,-,+,+,-,-,+,+,-,-,+,+,Origin at 222, at , , 0 from 1,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,P,4,mm (C,4v,No,. 99),1,Origin on 4mm,8,c,1,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,y,x,z,;,y,x,z,;,y,x,z,;,y,x,z,.,4,f,m,x,z; x,z; ,x,z; ,x,z.,4,e,m,x,0,z,;,x,0,z; 0,x,z; 0,x,z.,4,d,m,x,x,z,;,x,x,z,;,x,x,z,;,x,x,z,.,2,c,mm,0,z; 0,z.,1,b,4mm,z.,1,a,4mm,0,0,z,.,P,4nc,4mm Tetragonal,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,8,c,1,x,y,z,;,x,y,z,;,+x,-y,+z,;,-x,+y,+z,;,y,x,z,;,y,x,z,;,+y,+x,+z,;,-y,-x,-z,.,4,b,2,0,z,; ,0,z,;,0,+z;,0,+z.,2,a,4,0,0,z,; ,+z.,Origin on 4,6,C,4v,No.,104,P,4nc,4,d,1,x,y,z,;,x,y,z,;,y,x,z,;,y,x,z,.,2,c,2,0,z,; ,0,z,.,1,b,4,z.,1,a,4,0,0,z,.,P,4,P,mmm,(,D,2h,No,. 47),1,+,-,+,-,-,+,-,+,+,-,+,-,-,+,-,+,+,-,+,-,-,+,-,+,+,-,+,-,-,+,-,+,P,2/m,2/m,2/m,8,1,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,.,1,a,mmm,0,0,0,.,16,r,1,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,;,x,y,z,.,2,a,mmm,0,0,0,.,C,mmm,(,D,2h,No,. 65),19,Origin at center (,mmm,),特征,3,：初基,与,有心 等效位置数,(0,0,0; ,0) +,F,m3m (O,h,No,. 225),5,I,m3m (O,h,No,. 229),9,192,l,1,x,y,z,;,.,.,8,c,4,3m,; ,.,4,b,m3m, .,4,a,m3m,0,0,0.,(0,0,0; 0,; ,0,; ,0) +,96,l,1,x,y,z,;,.,.,8,c,3m,; ,; ,; ,.,6,b,4,/mmm,0,; ,0,; ,0.,2,a,m3m,0,0,0.,(0,0,0; ,) +,螺旋轴或滑移面不是,点式空间群,的,基本操作,。,至少有一个,基本操作,为,非点式操作,即为,非,点式空间群,点式空间群在单胞中一定至少有一个位置具有与空间群点群相同的,位置对称性,空间群操作：,r,= ,R,|,t,r,=,R,r,+,t,(,赛兹算符,),对非点式操作,t,=,，,是单胞的分数平移,对于点式操作,t,=,=,0,R,|,t,、,1,|,t,n,、,R,|,0,、,R,|,一般来说，对于给定的一组等效位置，,等效位置数,乘以,位置对称性,点群的阶,等于,空间群点群的阶。,对有心单胞，为,2h, 3h,或,4h,Orthorhombic,222,P,2,1,2,1,2,1,No.,19,Origin,halfway between three pairs of non-intersecting screw axes,+,+,+,_,+,+,+,_,_,P,2,1,2,1,2,1,D,2,4,Number of positions, Wyckoff notation, and point symmetry,Co-ordinates of equivalent positions,4,a,1,x,y,z,;,-x, y,+z,;,+x,-y, z,;,x,+y,-z,;,Orthorhombic,mmm,P,2,1,/,n,2,1,/,m,2,1,/,a,No.,62,Origin at 1,_,_,+,+,+,_,_,_,+,+,+,_,_,_,+,+,+,_,P,nma,D,2,h,16,Number of positions, Wyckoff notation, and point symmetry,Co-ordinates of equivalent positions,8,d,1,x,y,z,;,+x,-y,-z,;,x,+y,z,;,-x,y,+z,;,x,y,z,;,-x,+y,+z,;,x,-y,z,;,+x,y,-z,.,4,c,m,x,z; x,z; -x,+z; +x,-z.,4,b,1,0,0,; 0,; ,0,0; ,0.,4,a,1,0,0,0; 0,0; ,0,; ,.,P,2,1,2,1,2,1,P,2/m,2/m,2/m,Origin at 1,P,2,1,/,n,2,1,/,m,2,1,/,a,a,b,c,_,_,+,+,+,_,_,_,+,+,+,_,_,_,+,+,+,_,2,1,/,a,2,1,/,m,2,1,/,n,Number of positions, Wyckoff notation, and point symmetry,Co-ordinates of equivalent positions,_,_,+,+,+,_,_,_,+,+,+,_,_,_,+,+,+,_,P,nma,GdFeO,3,Gd,Fe,O,b,a,c,c,b,a,b,c,a,a,b,c,+,GdFeO,3,Gd,Fe,O,b,c,a,Gd,O,I,(4c),Fe (4a),O,II,(8d),x,?,P,m3m (O,h,No,. 221),1,a,b,c,48,n,1,x,y,z,;,.,.,3,c,4,/mmm,0,; ,0,; ,0.,1,b,m3m, .,1,a,m3m,0,0,0.,Sr,(,Ti,),Ti,(,Sr,),O,a, b, c,?,Sr,Ti,O,3,F,m3m (O,h,No,. 225),5,I,m3m (O,h,No,. 229),9,192,l,1,x,y,z,;,.,.,8,c,4,3m,; ,.,4,b,m3m, .,4,a,m3m,0,0,0.,(0,0,0; 0,; ,0,; ,0) +,96,l,1,x,y,z,;,.,.,8,c,3m,; ,; ,; ,.,6,b,4,/mmm,0,; ,0,; ,0.,2,a,m3m,0,0,0.,(0,0,0; ,) +,Ferromagnetism :,Superconductivity:,Ferroelectricity:,Multiferroics,(La,Sr)MnO,3,(La,Ca)MnO,3,SrRuO,3,YBa,2,Cu,3,O,7,(La,Sr)CuO,4,SrTiO,3,BaTiO,3,Pb(Zr,Ti)O,3,Pb(Mg1/3Nb2/3)O,3,BiFeO3,DyMnO3,Perovskite Structure,1,、,非点式空间群,举例分析,2,、,空间群国际表,举例分析,3,、,二维空间群,(,全部,),一维情况：,a,a,点阵，,Lattice,p,1,p,m,p,1,p,m,Oblique,a,b,90,o,Rectangular,a,b,=,90,o,Square,a,=,b,=,90,o,60,o,angle rhombus,Hexagonal,a,=,b,=,120,o,斜方,长方,有心长方,正方,六角,晶系,点群,布拉菲,点阵,73,种,点式空间群,三 斜,单 斜,正 交,四 方,三 方,六 方,立 方,P,P,P,P,P,P,P,1,1,m,2,2/m,222,mm2,mmm,4,2m,4,4,22,4,/mmm,4,mm,4,/m,4,3,m,3,3,m,3,2,3,6,22,6,/mmm,6,mm,6,/m,6,6,2m,6,2,3,m,3,4,3,2,m,3,m,4,3,m,P1,P1,Pm,P2,P2/m,P222,Pmm2,Pmmm,P,4,2m,P,4,P,4,22,P,4,/mmm,P,4,mm,P,4,/m,P,4,P3,1,m,P,3,P,3,m1,P,3,1,2,P,3,P2,3,Pm,3,P4,3,2,Pm,3,m,P4,3,m,Bm,B2,B2/m,C222,Cmm2,Cmmm,I222,Imm2,Immm,F222,Fmm2,Fmmm,Amm2,B,C,I,F,I,P4m2,I,4,2m,I,4,I,4,22,I,4,/mmm,I,4,mm,I,4,/m,I,4,I4m2,R,R,3,m,R,3,R,3,m,R,3,2,R,3,P321,P3m1,P31m,P,6,m2,P,6,P,6,22,P,6,/mmm,P,6,mm,P,6,/m,P,6,P62m,I,F,I2,3,Im,3,I4,3,2,Im,3,m,I4,3,m,F2,3,Fm,3,F4,3,2,Fm,3,m,F4,3,m,十七种,二维空间群,点阵,点群,空间群,序号,斜形,1,p,1,1,对应图像,p,矩形,p, c,正方形,p,六方形,p,2,p,2,11,p,4,4,3,3m,6,6mm,p,3,p,3m1,p,31m,p,6,p,6m,m,2,p,2,mm,p,2,m,g,P,2,gg,(n),c,2,mm,2mm,6,7,8,9,10,p,4m,m,C,4,g,m,(,g,),4mm,11,12,13,14,15,16,17,p,1,m,1,p,1,g,1,c,1,m,1,m,3,4,5,T,he,17,Two-dimensional Space Groups,E,quivalent positions,S,ymmetry, and,P,ossible reflections,p,1,p,2,p,2,11,1,a,1,x,y,.,Origin on 1,Number of positions, Wyckoff notation, and point symmetry,Origin at 2,2,e,1,x, y, x, y.,1,d,2,1/2, 1/2.,1,c,2,1/2, 0.,1,b,2,0,1/2.,1,a,2,0,0.,p,m,p,1,m,1,p,g,p,1,g,1,c,m,c,1,m,1,Origin on m,2,c,1,x, y;,x,y,.,1,b,m,1/2, y.,1,a,m,0, y.,Origin on m,4,b,1,x, y; x, y.,2,a,m,0, y.,Co-ordinates of equivalent positions,(0,0; 1/2,1/2)+,Origin on g,2,a,1,x, y; x,1/2+y.,p,mm,p,2,mm,p,m,g,p,2,m,g,Origin at 2mm,Origin at 2,4,i,1,x,y,;,x,y,;,x,y,;,x,y,.,2,h,m,y;,y.,2,g,m,0,y; 0,y.,2,f,m,x,1/2; x,1/2.,2,e,m,x,0; x,0.,1,d,mm,1/2,1/2.,1,c,mm,1/2,0.,1,b,mm,0,1/2.,1,a,mm,0, 0.,4,d,1,x,y,;,x,y,; +x,y; -x,y.,2,c,m,;y; ,y.,2,b,2,0,; ,.,2,a,2,0,0; ,0.,p,gg,p,2,gg,Origin at 2,4,c,1,x,y,;,x,y,; +x,-y; -x,+y.,2,b,2,0;,0,.,2,a,2,0,0; , .,c,mm,c,2,mm,Origin at 2mm,Co-ordinates of equivalent positions,(0,0; 1/2,1/2)+,8,f,1,x,y,;,x,y,;,x,y,;,x,y,.,4,e,m,0,y; 0,y.,4,d,m,x,0; x,0.,4,c,2,; ,.,2,b,mm,0,.,2,a,mm,0,0.,p,4,p,4,p,4m,p,4m,m,Origin at 4,4,d,1,x,y,;,x,y,;,y,x,;,y,x,.,2,c,2,0; 0,.,1,b,4,.,1,a,4,0,0.,Origin at 4mm,8,g,1,x,y,;,x,y,;,y,x,;,y,x,;,x,y,;,x,y,;,y,x,;,y,x,.,4,f,m,x,x,;,x,x,;,x,x,;,x,x,.,4,e,m,x,; x,; ,x; ,x.,4,d,m,x,0; x,0; 0,x; 0,x.,2,c,mm,0; 0,.,1,b,4mm,.,1,a,4mm,0,0.,a,b,c,P,4mm,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,P,4bm,p,4,g,p,4,g,m,Origin at 4,8,d,1,x,y,;,y,x,; -x,+y; -y,-x;,x,y,;,y,x,; +x,-y; +y,+x.,4,c,m,x,+x; x,-x; +x,x; -x,x.,2,b,mm,2,0; 0,.,2,a,4,0,0;,.,P,4bm,Plus,“+, and,z”,?,p,3,p,3,Origin at 3,3,d,1,x,y,;,y,x-y,;,y-x,x,.,1,c,3,2/3,1/3,.,1,b,3,1/3,2/3,.,1,a,3,0,0,.,p,3m1,p,3m1,p,31m,p,31m,Origin at 3m1,Origin at 31m,6,e,1,x,y,;,y,x-y,;,y-x,x,;,y,x,;,x,x-y,;,y-x,y,.,3,d,m,x,x,;,x,2x,;,2x,x,.,1,c,3m,2/3,1/3,.,1,b,3m,1/3,2/3,.,1,a,3m,0,0,.,6,d,1,x,y,;,y,x-y,;,y-x,x,;,y,x,;,x,y-x,;,x-y,y,.,3,c,m,x,0; 0,x;,x,x,.,2,b,3,1/3,2/3,;,2/3,1/3,1,a,3m,0,0,.,p,6,p,6,6,d,1,x,y,;,y,x-y,;,y-x,x,;,x,y,;,y,y-x,;,x-y,x,.,3,c,2,0,; 0,; ,.,2,b,3,1/3,2/3,;,2/3,1/3,.,1,a,6,0, 0.,Origin at 6,p,6m,p,6m,m,Origin at 6mm,12,f,1,x,y,;,y,x-y,;,y-x,x,;,y,x,;,x,y-x,;,x-y,y,;,x,y,;,y,y-x,;,x-y,x,;,y,x,;,x,x-y,;,y-x,y,.,6,e,m,x,x,;,x,2x,;,2x,x,;,x,x,; x,2x; 2x,x.,6,d,m,x,0,;,0,x,;,x,x,;,x,0; 0,x;,x,x,.,3,c,mm,1/2,0,;,0,1/2,;,1/2,1/2,.,2,b,3m,1/3,2/3,;,2/3, 1/3.,1,a,6mm,0,0,.,p,3m1,p,3m1,作业：,作下面空间群的俯视图（一般等效位置和对称操作）：用赛兹算符推导；并给出位置数、,Wyckoff,表示、位置对称性和等效位置：,p6,1,p4,3,I4, p4,mm,